The only help I can give you is inspiration from my own world building. I am a terrible worldbuilder and use the map of Eurasia and Northern Africa as a rough placement for where tribes come from. So I am just pulling languages from where I have the largest concentration of that particular people.
Dwarvish: Icelandic. Modern speakers of Icelandic can read the Prose Eddas as written just fine, and it makes sense for a tradition bound, long lived race to have a language that barely changes over the course of generations.
Elvish: I would say Irish (Gaelige), as it's a Celtic language, but is more flowing in my experience than Scottish Gaelic.
Giant: Greek
Gnomish/Halfling: No idea. I would just give them an Old English/Finnish dialect and say that they have lost their old tongue due to various diasporas and having to fit in with the current regime of the area.
Goblin: Sanskrit. The hierarchy between the subraces that keeps people in their place really invokes the idea of the historical Indian caste system
Orc: Russian/Mongolian (I place Orcs on the mongolian steppes and farther North
Celestial: Hebrew (I think they already use the Hebrew alphabet for the language, and it will likely have some resonance with the table. Perhaps not, I am not sure what your table's likely experience with Judeo-Christian themes.
Draconic: German. It fits well with the fact that Kobold means "Goblin" in German.
Infernal: Latin. It sounds so grandiose and evil in English
Abyssal:Frenchno good ideas
Primordial: no idea. Perhaps Esperanto if you want to do a "First language" idea, since it is supposed to be one of the easiest languages to learn.
Undercommon: No really good ideas. Is there a language that shares common ties to Finnish that has diverged since? Might be the best way to do it.
Sylvan: Perhaps use Scottish Gaelic here for ties between Elven and Sylvan.
Druidic: No good ideas. If you have a better idea for Primordial, perhaps a similar language for Druidic would work.
Thieve's Cant: I would use Cockney Rhyming Slang personally, but that isn't in Google Translate.
This is really surface level breakdown. Someone else probably has much better ideas than my list. Mine is based a lot off of laziness in world building, so take it with a grain of salt.